Question 1:
a) A simple harmonic oscillator has amplitude of 0.17 m and a period of 0.84 s. Find out the frequency and the angular frequency of the motion. Write down expressions for the time dependence of displacement, velocity and acceleration. What are the maximum values of the velocity and acceleration?
b) Assume that a spring-mass system has k = 18 Nm-1 and m = 0.71 kg. The system is oscillating with amplitude of 54 mm.
i) Find out the angular frequency of oscillation.
ii) Obtain an expression for the velocity v of the block as a function of displacement, x and compute v at x = 34 mm
iii) Obtain an expression for the mass’s distance |x| from the equilibrium position as a function of the velocity v and compute | x | when v = 0.18 ms−1
iv) Compute the energy of the spring-mass system.
Question 2:
a) Establish the equation of motion of a damped oscillator. Show that for a weakly damped oscillator, displacement is given by:
x(t) = ao exp (-bt) sinωdt
As well obtain the expression for the time period of oscillation.
b) The weight on a vertical spring undergoes forced vibrations according to the given equation:
d2x/dt2 + 4x = 8 sin ωt
Where x is the displacement from the equilibrium position and ω > 0 is a constant. If at t = 0, x = 0 and v (= dx/dt) = 0; Compute
i) x as a function of t, and
ii) The period of the external force for which resonance takes place.
Question 3:
a) A sinusoidal wave is illustrated by:
y (x, t) = 2.0 sin (2.11 x – 3.62 t)
Where x is the position all along the wave propagation. Find out the amplitude, wave number, wavelength, frequency and velocity of the wave.
b) The linear density of a vibrating string is 1.3 × 10−4 kg m−1. A transverse wave is propagating on the string and is illustrated by the equation:
y (x, t) = 0.021 sin (x − 30t)
Where x and y are in meters and t is in seconds. Compute the tension in the string.
c) A transverse wave of amplitude 0.01 m is produced at one end (x = 0) of a long horizontal string by a turning fork of frequency 500 Hz. At a given instant of time, the displacement of the particle at x = 0.1 m is − 0.005 m and that of the particle at x = 0.2 m is 0.005 m. Compute the wavelength and the wave velocity.
Question 4:
a) A travelling wave is illustrated by the equation:
y = exp (− az2 – bt2 − 2 √ab zt)
Compute the wave velocity and sketch the waveform at t = 0 and t = 3s. Take a = 144 cm−2 and b = 9 s−1.
b) A stretched string of mass 20 g vibrates with a frequency of 30 Hz in its basic mode and the supports are 40 cm apart. The amplitude of vibrations at antinode is 4 cm. Compute the velocity of propagation of the wave in the string and also the tension in it.
c) Consider two cylindrical pipes of equivalent length. One of such acts as a closed organ pipe and the other as open organ pipe. The frequency of the second harmonic in the closed pipe is 200 Hz higher than the first harmonic of the open pipe. Compute the fundamental frequency of the closed pipe.